There's many many ways to solve this of course. Another possible solution might have been something that looks a bit like this:

Leave it on his primary server and use cpulimit to constrain CPU usage of ffmpeg to, say, 30% - this addresses the throttling problem:

cpulimit -l 30 -- ffmpeg -i input.mp4 -c:v libx264 -preset veryfast output.mp4

Then use the simplest of queueing mechanisms which is just to move files between directories - a long proven and reliable way to do a processing queue:

bash code here:

    #!/usr/bin/env bash
    set -u
    INCOMING="./incoming"
    PROCESSING="./processing"
    DONE="./done"
    FAILED="./failed"
    CPU_LIMIT=30
    SLEEP_SECONDS=2
    mkdir -p "$INCOMING" "$PROCESSING" "$DONE" "$FAILED"
    # recover files left in processing after a crash
    for path in "$PROCESSING"/*; do
        [ -e "$path" ] || continue
        name="$(basename "$path")"
        mv "$path" "$INCOMING/$name"
    done
    while true; do
        FILE=""
        # find first file in incoming
        for path in "$INCOMING"/*; do
            [ -f "$path" ] || continue
            FILE="$path"
            break
        done
        # if nothing available wait
        if [ -z "$FILE" ]; then
            sleep "$SLEEP_SECONDS"
            continue
        fi
        BASENAME="$(basename "$FILE")"
        STEM="${BASENAME%.*}"
        WORK="$PROCESSING/$BASENAME"
        OUTPUT="$DONE/$STEM.mp4"
        # move file into processing
        mv "$FILE" "$WORK"
        echo "Processing $WORK"
        # run ffmpeg under cpu limit (blocking)
        cpulimit -l "$CPU_LIMIT" -- ffmpeg -y -threads 1 -i "$WORK" -c:v libx264 -preset veryfast -c:a aac "$OUTPUT"
        STATUS=$?
        if [ "$STATUS" -eq 0 ]; then
            rm "$WORK"
            echo "Finished $OUTPUT"
        else
            mv "$WORK" "$FAILED/$BASENAME"
            echo "Failed $FAILED/$BASENAME"
        fi
    done